Textures Not Showing in Viewport (Blender 2.8)
Blender 2.8 uses an entirely different materials system from previous versions. As a consequence of this (and perhaps also as a result of some missing backwards compatibility features in Blender 2.8), image-based materials created in older versions of blender get replaced with plain, solid-color materials with no textures when opened in Blender 2.8.

To fix this, you will need to manually re-assign textures to their appropriate materials in the project. Select the model you want to fix, then in the properties editor go to the materials tab, select a material to fix, and click "Use Nodes". Next, click the small button to the right of the "Base Color", and click "Image Texture". Finally, click the image dropdown below the base color, and select the correct texture for the material you have selected. Depending on how well named the materials and textures are, you may have to use a bit of trial and error to figure out which texture goes where. Now repeat this process for every material in the project.

Yes, this could be extremely tedious depending on the size of your project, but until Blender introduces a built-in way to convert old models to the new materials system this is what we're stuck with, apparently.

For more information on this issue, see Opening old .blend files (missing images) on KatsBits.